le.eq {SNSequate}

Local equating methods

Description

This function implements the local method of equating as descibed in van der Linden (2011).

Usage

le.eq(S.X, It.X, It.Y, Theta)

Arguments

S.X

A vector containing the observed scores of the sample taking test X.

It.X

A matrix of item parameter estimates coming from an IRT model for test form X (difficulty, discrimation and guessing parameters are located in the first, second and third column, respectively).

It.Y

A matrix of item parameter estimates coming from an IRT model for test form Y.

Theta

Either a number or vector of values representing the value of theta where to condition on (see details)

Details

The function implements the local equating method as described in van der Linden (2011). Based on Lord (1980) principle of equity, local equating methods utilizes the conditional on abilities distributions of scores to obtain the transformation \varphi. The method leads to a family of transformations of the form

\varphi(x;\theta)=G_{Y\mid\theta}^{-1}(F_{X\mid\theta}(x)),\quad \theta\in\mathcal{R}

The conditional distributions of X and Y are obtained using the algorithm described by Lord and Wingersky (1984). Among other possibilities, a value for \theta can be a EAP, ML or MAP estimation of it, for and underlying IRT model (for example, using the ltm R package (Rizopoulos, 2006)).

Value

A list containing the observed scores to be equated, the corresponding ability estimates where to condition on, and the equated values

Author(s)

Jorge Gonzalez [e-mail]

References

Gonzalez, J. (2014). SNSequate: Standard and Nonstandard Statistical Models and Methods for Test Equating. Journal of Statistical Software, 59(7), 1-30.

Lord, F. (1980). Applications of Item Response Theory to Practical Testing Problems. Lawrence Erlbaum Associates, Hillsdale, NJ.

Lord, F. and Wingersky, M. (1984). Comparison of IRT True-Score and Equipercentile Observed-Score Equatings. Applied Psychological Measurement,8(4), 453–461.

Rizopoulos, D. (2006). ltm: An R package for latent variable modeling and item response theory analyses. Journal of Statistical Software, 17(5), 1–25.

van der Linden, W. (2011). Local Observed-Score Equating. In A. von Davier (Ed.) Statistical Models for Test Equating, Scaling, and Linking. New York, NY: Springer-Verlag.

See Also

mea.eq, eqp.eq, lin.eq ker.eq

Examples

## Artificial data for two 5-items tests forms. Both forms are assumed
## being fitted by a 3PL model.

## Create (artificial) item parameters matrices for test form X and Y
ai<-c(1,0.8,1.2,1.1,0.9)
bi<-c(-2,-1,0,1,2)
ci<-c(0.1,0.15,0.05,0.1,0.2)
itx<-rbind(bi,ai,ci)
ai<-c(0.5,1.4,1.2,0.8,1)
bi<-c(-1,-0.5,1,1.5,0)
ci<-c(0.1,0.2,0.1,0.15,0.1)
ity<-rbind(bi,ai,ci)

#Two individuals with different ability (1 and 2) obtain the same score 2.
#Their corresponding equated scores values are:
le.eq(c(2,2),itx,ity,c(1,2))